How to evaluate the matrix element of coulomb repulsion term between electrons in an atom suing spherical harmonics multipole expansion? This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf

I am trying to evaluate the two electron matrix element between $$\langle2p_{1}(\vec{r}_{1})2p_{-1}(\vec{r}_{2})|\frac{1}{|\vec{r}_{1}-\vec{r}_{2}|}|2p_{1}(\vec{r}_{1})2p_{-1}(\vec{r}_{2})\rangle$$ for a two electron atom using the relation attached in the image.I am trying to apply the formula for this case where $$m_{i}=1,m_{i'}=-1,m_{j}=1,m_{j'}=-1$$ and $$l_{i}=l_{i'}=l_{j}=l_{j'}=1$$

I really dont understand how to apply in the given formula as the summation is over all $$m$$'s. however, in the derivation it was clear that subscripted $$l$$'s and $$m$$'s are fixed and the one without subscript is varying. How do I apply this formula to evaluate these matrix elements ? Can someone suggest a better reference and help me in applying this formula

Source for this pdf :http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf

$$\frac{1}{r_{12}} = 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l\frac{1}{2l+1}\frac{r_<^l}{r_>^{l+1}}Y^*_{lm}(\hat{r}_2)Y_{lm}(\hat{r}_1)$$